# On the universal p-adic sigma and Weierstrass zeta functions

## TL;DR

This paper introduces a new derivation method for universal p-adic sigma and Weierstrass zeta functions, emphasizing coefficient congruences and applicable to generalized elliptic curves.

## Contribution

It provides a novel derivation approach for p-adic elliptic functions that works uniformly for ordinary and generalized elliptic curves.

## Key findings

- New derivation method for p-adic sigma and zeta functions
- Highlights congruences among Laurent expansion coefficients
- Applicable to generalized elliptic curves

## Abstract

For primes $p>3$ we produce a new derivation of the universal $p$-adic sigma function and $p$-adic Weierstrass zeta functions of Mazur and Tate for ordinary elliptic curves by a method that highlights congruences among coefficients in Laurent expansions of elliptic functions, and works simultaneously for generalized elliptic curves defined by Weierstrass equations.